标题： 全9轮和减少轮数的库兹涅齐克密码的扩展c-差分区分器 显示英文标题

标题： Extended c-differential distinguishers of full 9 and reduced-round Kuznyechik cipher

摘要： 本文介绍了{\em 截断内部$c$-差分密码分析}，一种首次使$c$-微分均匀性在分组密码中实际应用成为可能的新技术。虽然 Ellingsen 等人（IEEE Trans. Inf. Theory, 2020）使用$(F(x\oplus a), cF(x))$建立了$c$-微分均匀性的概念，但一个关键挑战依然存在：乘以$c$会破坏分组密码分析中关键加法所必需的结构特性。 我们通过开发一个\emph{内部的} $c$ -微分方法来解决这个挑战，其中乘以$c$影响输入：$(F(cx\oplus a), F(x))$。 我们证明了函数$F$的内部$c$-微分均匀性等于$F^{-1}$的外部$c$-微分均匀性，从而建立了基本的对偶性。 这种修改保留了密码结构，同时实现了实际的密码分析应用。 我们的主要贡献是一个全面的多方面统计计算框架，针对完整的9轮Kuznyechik密码实施截断的$c$-微分分析（内部$c$-差异对后端密钥异或具有免疫性）。 通过涉及数百万个微分对的广泛计算分析，我们在所有测试的轮数中都展示了统计上显著的非随机性。 对于完整的9轮密码，我们识别出多个触发关键安全警报的配置，偏差比达到$1.7\times$，校正后的p值低至$1.85 \times 10^{-3}$，这表明对这种新的攻击向量的安全性余量不足。 这是针对完整9轮Kuznyechik的第一个实际区分器。 显示英文摘要

摘要： This paper introduces {\em truncated inner $c$-differential cryptanalysis}, a novel technique that for the first time enables the practical application of $c$-differential uniformity to block ciphers. While Ellingsen et al. (IEEE Trans. Inf. Theory, 2020) established the notion of $c$-differential uniformity using $(F(x\oplus a), cF(x))$, a key challenge remained: multiplication by $c$ disrupts the structural properties essential for block cipher analysis, particularly key addition. We resolve this challenge by developing an \emph{inner} $c$-differential approach where multiplication by $c$ affects the input: $(F(cx\oplus a), F(x))$. We prove that the inner $c$-differential uniformity of a function $F$ equals the outer $c$-differential uniformity of $F^{-1}$, establishing a fundamental duality. This modification preserves cipher structure while enabling practical cryptanalytic applications. Our main contribution is a comprehensive multi-faceted statistical-computational framework, implementing truncated $c$-differential analysis against the full 9-round Kuznyechik cipher (the inner $c$-differentials are immune to the key whitening at the backend). Through extensive computational analysis involving millions of differential pairs, we demonstrate statistically significant non-randomness across all tested round counts. For the full 9-round cipher, we identify multiple configurations triggering critical security alerts, with bias ratios reaching $1.7\times$ and corrected p-values as low as $1.85 \times 10^{-3}$, suggesting insufficient security margin against this new attack vector. This represents the first practical distinguisher against the full 9-round Kuznyechik.